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Theorem elsnxp 6290
Description: Membership in a Cartesian product with a singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.) (Proof shortened by JJ, 14-Jul-2021.)
Assertion
Ref Expression
elsnxp (𝑋𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩))
Distinct variable groups:   𝑦,𝐴   𝑦,𝑉   𝑦,𝑋   𝑦,𝑍

Proof of Theorem elsnxp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elxp 5699 . . 3 (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑥𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)))
2 df-rex 3071 . . . . . 6 (∃𝑦𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
3 an13 645 . . . . . . 7 ((𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) ↔ (𝑦𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
43exbii 1850 . . . . . 6 (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
52, 4bitr4i 277 . . . . 5 (∃𝑦𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)))
6 elsni 4645 . . . . . . . . 9 (𝑥 ∈ {𝑋} → 𝑥 = 𝑋)
76opeq1d 4879 . . . . . . . 8 (𝑥 ∈ {𝑋} → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
87eqeq2d 2743 . . . . . . 7 (𝑥 ∈ {𝑋} → (𝑍 = ⟨𝑥, 𝑦⟩ ↔ 𝑍 = ⟨𝑋, 𝑦⟩))
98biimpa 477 . . . . . 6 ((𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
109reximi 3084 . . . . 5 (∃𝑦𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
115, 10sylbir 234 . . . 4 (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
1211exlimiv 1933 . . 3 (∃𝑥𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
131, 12sylbi 216 . 2 (𝑍 ∈ ({𝑋} × 𝐴) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
14 snidg 4662 . . . . 5 (𝑋𝑉𝑋 ∈ {𝑋})
15 opelxpi 5713 . . . . 5 ((𝑋 ∈ {𝑋} ∧ 𝑦𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
1614, 15sylan 580 . . . 4 ((𝑋𝑉𝑦𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
17 eleq1 2821 . . . 4 (𝑍 = ⟨𝑋, 𝑦⟩ → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴)))
1816, 17syl5ibrcom 246 . . 3 ((𝑋𝑉𝑦𝐴) → (𝑍 = ⟨𝑋, 𝑦⟩ → 𝑍 ∈ ({𝑋} × 𝐴)))
1918rexlimdva 3155 . 2 (𝑋𝑉 → (∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩ → 𝑍 ∈ ({𝑋} × 𝐴)))
2013, 19impbid2 225 1 (𝑋𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wrex 3070  {csn 4628  cop 4634   × cxp 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-opab 5211  df-xp 5682
This theorem is referenced by:  esum2dlem  33085  esum2d  33086  projf1o  43886  sge0xp  45135
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